Semigroup Forum

, Volume 89, Issue 1, pp 77–104 | Cite as

On semigroups of endomorphisms of a chain with restricted range

  • Vítor H. Fernandes
  • Preeyanuch Honyam
  • Teresa M. Quinteiro
  • Boorapa Singha


Let X be a finite or infinite chain and let \({\mathcal{O}}(X)\) be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of \({\mathcal{O}}(X)\) and Green’s relations on \({\mathcal{O}}(X)\). In fact, more generally, if Y is a nonempty subset of X and \({\mathcal{O}}(X,Y)\) is the subsemigroup of \({\mathcal{O}}(X)\) of all elements with range contained in Y, we characterize the largest regular subsemigroup of \({\mathcal{O}}(X,Y)\) and Green’s relations on \({\mathcal{O}}(X,Y)\). Moreover, for finite chains, we determine when two semigroups of the type \({\mathcal {O}}(X,Y)\) are isomorphic and calculate their ranks.


Transformations Order-preserving Restricted range Rank 



This research was mainly carried out during the visit of the second and fourth authors to Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa and Centro de Álgebra da Universidade de Lisboa between August and October 2011.

The authors would like to thank Cláudia and Francisco Coelho for their help in reviewing the text of this paper.


  1. 1.
    Adams, M.E., Gould, M.: Posets whose monoids of order-preserving maps are regular. Order 6, 195–201 (1989) zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Aĭzenštat, A.Ya.: The defining relations of the endomorphism semigroup of a finite linearly ordered set. Sib. Mat. Zh. 3, 161–169 (1962) (Russian) Google Scholar
  3. 3.
    Aĭzenštat, A.Ya.: Homomorphisms of semigroups of endomorphisms of ordered sets. Uč. Zap.—Leningr. Pedagog. Inst. 238, 38–48 (1962) (Russian) Google Scholar
  4. 4.
    Aĭzenštat, A.Ya.: Regular semigroups of endomorphisms of ordered sets. Uč. Zap.—Leningr. Gos. Pedagog. Inst. 387, 3–11 (1968) (Russian) Google Scholar
  5. 5.
    Almeida, J., Volkov, M.V.: The gap between partial and full. Int. J. Algebra Comput. 8, 399–430 (1998) zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Fernandes, V.H.: Semigroups of order-preserving mappings on a finite chain: a new class of divisors. Semigroup Forum 54, 230–236 (1997) zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Fernandes, V.H.: Semigroups of order-preserving mappings on a finite chain: another class of divisors. Russ. Math. (Izv. VUZ) 3(478), 51–59 (2002) (Russian) MathSciNetGoogle Scholar
  8. 8.
    Fernandes, V.H., Sanwong, J.: On the rank of semigroups of transformations on a finite set with restricted range. Algebra Colloq. (to appear) Google Scholar
  9. 9.
    Fernandes, V.H., Volkov, M.V.: On divisors of semigroups of order-preserving mappings of a finite chain. Semigroup Forum 81, 551–554 (2010) zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fernandes, V.H., Jesus, M.M., Maltcev, V., Mitchell, J.D.: Endomorphisms of the semigroup of order-preserving mappings. Semigroup Forum 81, 277–285 (2010) zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Gomes, G.M.S., Howie, J.M.: On the ranks of certain semigroups of order-preserving transformations. Semigroup Forum 45(3), 272–282 (1992) zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hall, M. Jr.: Combinatorial Theory. Wiley, New York (1967) zbMATHGoogle Scholar
  13. 13.
    Higgins, P.M.: Divisors of semigroups of order-preserving mappings on a finite chain. Int. J. Algebra Comput. 5, 725–742 (1995) zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Howie, J.M.: Products of idempotents in certain semigroups of transformations. Proc. Edinb. Math. Soc. 17, 223–236 (1971) zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, Oxford (1995) zbMATHGoogle Scholar
  16. 16.
    Jitjankarn, P.: Isomorphism Theorems for Semigroups of Order-preserving Full Transformations (2012). arXiv:1202.2977v1 [math.RA]
  17. 17.
    Kemprasit, Y., Changphas, T.: Regular order-preserving transformation semigroups. Bull. Aust. Math. Soc. 62(3), 511–524 (2000) zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kim, V.I., Kozhukhov, I.B.: Regularity conditions for semigroups of isotone transformations of countable chains. Fundam. Prikl. Mat. 12(8), 97–104 (2006) (Russian). Translation in J. Math. Sci. (N.Y.) 152(2), 203–208 (2008) Google Scholar
  19. 19.
    Mendes-Gonçalves, S., Sullivan, R.P.: The ideal structure of semigroups of transformations with restricted range. Bull. Aust. Math. Soc. 83, 289–300 (2011) zbMATHMathSciNetGoogle Scholar
  20. 20.
    Mora, W., Kemprasit, Y.: Regular elements of some order-preserving transformation semigroups. Int. J. Algebra 4(13–16), 631–641 (2010) zbMATHMathSciNetGoogle Scholar
  21. 21.
    Nenthein, S., Youngkhong, P., Kemprasit, Y.: Regular elements of some transformation semigroups. Pure Math. Appl. 16(3), 307–314 (2005) zbMATHMathSciNetGoogle Scholar
  22. 22.
    Repnitskiĭ, V.B., Vernitskii, A.: Semigroups of order preserving mappings. Commun. Algebra 28(8), 3635–3641 (2000) zbMATHCrossRefGoogle Scholar
  23. 23.
    Repnitskiĭ, V.B., Volkov, M.V.: The finite basis problem for the pseudovariety \(\mathcal{O}\). Proc. R. Soc. Edinb., Sect. A, Math. 128, 661–669 (1998) zbMATHCrossRefGoogle Scholar
  24. 24.
    Sanwong, J., Sommanee, W.: Regularity and Green’s relations on a semigroup of transformations with restricted range. Int. J. Math. Math. Sci. (2008). Art. ID 794013, 11 pp. Google Scholar
  25. 25.
    Sanwong, J., Singha, B., Sullivan, R.P.: Maximal and minimal congruences on some semigroups. Acta Math. Sin. Engl. Ser. 25(3), 455–466 (2009) zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Sullivan, R.P.: Semigroups of linear transformations with restricted range. Bull. Aust. Math. Soc. 77, 441–453 (2008) zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Symons, J.S.V.: Some results concerning a transformation semigroup. J. Aust. Math. Soc. A 19, 413–425 (1975) zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Vernitskii, A., Volkov, M.V.: A proof and generalisation of Higgins’ division theorem for semigroups of order-preserving mappings. Izv. Math. Izv. Vuzov Matematika 1, 38–44 (1995) (Russian) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Vítor H. Fernandes
    • 1
    • 2
  • Preeyanuch Honyam
    • 3
  • Teresa M. Quinteiro
    • 2
    • 4
  • Boorapa Singha
    • 5
  1. 1.Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaCaparicaPortugal
  2. 2.Centro de Álgebra da Universidade de LisboaLisboaPortugal
  3. 3.Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand
  4. 4.Instituto Superior de Engenharia de LisboaLisboaPortugal
  5. 5.School of Mathematics and Statistics, Faculty of Science and TechnologyChiang Mai Rajabhat UniversityChiang MaiThailand

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